C₅×#8816 #11931536

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₅×#8816

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

422 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5)(6,7,8,9)(11,12)(13,14)(16,17)(18,19)(20,26)(21,29)(22,24)(23,27)(25,30)(28,32)(31,33);(10,11,14,16,15,17,13,12)(18,20,21,27,22,28,30,33)(19,23,25,26,24,31,29,32);(10,12,13,17,15,16,14,11)(19,24)(23,31)(25,29)(26,32)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1804);